Anisotropy in tensile properties of plain weave fabric

Abstract

Tensile properties and geometrical behavior of a plain weave fabric under arbitrary direction are addressed in this study. A geometrical–mechanical model is established on the basis of the deflection curve and flattening of tensioned yarn. For this purpose, a differential equation to assess the curvature of tensioned yarn under a certain bending moment is proposed. Different conditions of a fabric, namely relaxed fabric, stretched in principal directions and arbitrary directions are discussed. Ordinary data of a fabric and constituted yarns are input data of this model. The tensile-force curve and geometry of a strip fabric under stress in the arbitrary direction can be estimated by the proposed model simultaneously.

Keywords

mechanical properties of woven fabric fabric geometry yarn and fabric deformation

Nomenclature

Parameter	Unit	Comment
ρ	gr/cm³	Density of fibers
$ρ_{(x, y)}$	m	Curvature radius at point (x,y)
$ɛ_{y 1}$	%	Warp yarn strain
$ɛ_{y 2}$	%	Weft yarn strain
β₁,β₂	Radian	Inflection angle of warp and weft
μ₁,μ₂	–	Packing density of warp and weft
\|d_f\|/2	m	Distance between warp/weft and neutral plane of fabric
C₁, C₂	–	Warp and weft crimp
d0₁, d0₂	m	Warp, weft yarn original diameter before weaving
D₁, D₂	Ends/m	Warp and weft yarn sett
e₁, e₂	–	Strain of fabric in warp and weft directions
E₁, E₂	N	Modulus of elasticity of warp, weft yarns
e_d	–	Flattening coefficient
e_f	%	Fabric strain in pure shear area
F_n	N	Normal force on cross-over point
G₁(F_n), G₂(F_n)	–	Flattening deformation of yarns as a function of normal force for warp, weft
h₁, h₂	m	Height of warp or weft yarn path
L₁, L₂	m	Length of warp and weft in unit cell of the structure
M ₍ _x _, _y ₎	Nċm	Bending moment at point (x,y)
N	–	Number of unit cells in width of ω
S ₍ _x _, _y ₎	Nċm²	Stiffness of yarn at point (x,y)
T₁, T₂	N	Planar force applied on one warp or weft yarn
Tex₁, Tex₂	tex	Linear density of warp and weft
α	Radian	Inclination angle of square in center of sample
φ	Radian	Initial angle of force axis with weft yarns
ω	m	Width of specimen
Ψ₁, Ψ₂	Radian	Warp, weft yarn angles with applied tension axis

Introduction

The properties of a woven fabric are not only important in principal directions, but also the behavior of woven fabric in other directions plays a vital role in some fabric applications. The behavior of fabric under stress in principal directions or simple shear deformation has been investigated by a number of researchers. However, the deformation and tensile properties of woven fabric are extremely anisotropic. In order to obtain excellent overall mechanical properties of applications, understanding the anisotropy of woven fabric properties is necessary. For instance, in the finite element method (FEM) isotropic materials are usually assumed for woven fabric, which conduct some errors during simulation. Estimating the tensile properties of fabric in an arbitrary direction will considerably increase the accuracy of the FEM.

To interpret the behavior of woven fabric in the bias direction, the concept of Weissenberg¹ and Weissenberg et al.² has been contemplated by a number of researchers. On the basis of this theory it is assumed that the motion of yarns at the interlacing point is pin-jointed motion and the incompressible and inextensible warp and weft yarns in a unit cell of fabric have constant yarn density. Although Weissenberg’s assumptions are acceptable in ordinary fabrics under low shear deformation, the dimensions of yarns and yarn setts are subjected to change in high shear deformation. Lindberg et al.³ dealt with the shear behavior of woven fabric. They found that shear deformation takes place on the basis of two levels: initial shear without sliding and shear with sliding, and then jamming. They mentioned that shear deformation is initially resisted by friction in the cross-over force and then in elastic forces after jamming. Kilby⁴ suggested a formula for calculating Young’s modulus of the fabric in the arbitrary direction on the basis of Young’s modulus of fabric in the warp and weft directions. To interpret the behavior of fabric in low stress deformation, Olofsson⁵ proposed a model by considering the yarn geometry as a function of external forces and reaction forces in the fabric and assuming a relation between the curvature of the yarn in the fabric and in the released state. Moreover, to compare various models, some investigations have been performed.^6,7 Treloar⁸ demonstrated that the test sample dimension has a significant effect on the behavior of fabric in the simple shear. He mentioned that the maximum shear strain that can be applied without the occurrence of wrinkling is dependent on the shape of the specimen and the length–width ratio. Later on, several distinct modes of deformation were put forward by Grosberg and Park,⁹ depending on the degree of shear imposed upon the fabric: (1) deformation due to rigid intersections when the shear is too small to overcome the friction; (2) yarn slippage at the intersection; (3) an elastic deformation when slipping is complete; and (4) jamming in the structure. Accordingly they deducted a model on the basis of the mechanical properties of yarns and the geometry of fabric structure for predicting initial shearing characteristics.

Soon after, Grosberg and Park¹⁰ suggested a model to predict the mechanism of deformation of woven fabric under shear stress in the elastic region. Spivak and Treloar¹¹ tried to find some relationship between bias extension and simple shear test methods. In this study, it is declared that the distribution of stress in the simple shear and bias extension test is not the same, and they mentioned that the rupture mechanism in the bias extension test is strongly dependent on the shape and dimension of the specimen. Meanwhile, the stress distribution across the specimen in the bias extension test would be a maximum at the center, then falling off to zero at the edges. They pointed out that the mechanism of rupture has been gliding to disassembly in edges, that is, the frictional forces at the cross-over were not sufficient to maintain the pin-pointing at the cross-over points. Skelton¹² studied the limitation in shear deformation of a fabric. In this study it is pointed out that the limits of shear are usually defined geometrically. For a wide range of conventional fabrics the shear limit is essentially defined by the side-by-side contact of one set of yarns. Leaf and Sheta¹³ demonstrated that the Young’s modulus of a fabric under bias stress is connected with the shear modulus and fabric Poisson’s ratio in the warp and weft directions.

Ning and Mee-Young¹⁴ tried to show anisotropy in the strength of fabric. In their study the anisotropy in the strength of fabric was plotted by applying the harmonic cosine series on measured values of strength as a pretest. Clearly, the accuracy of this method depends on the number of pretests in different directions. Buckenham¹⁵ compared the simple shear and bias extension test method again after Spivak and Treloar.¹¹ He mentioned that the bias extension test is more appropriate for industrial use than simple shear. Prodromou and Chen¹⁶ tried to find a relationship between shear angles and wrinkling of several layers of woven fabrics that were used for composite performances. They measured the angular deformation of warp and weft as shear angles by usage a trellis frame with the image processing method. Moreover, buckling or wrinkling that occurred due to compressive forces was studied in this research. Critical shear angle, the so-called ‘locking angle’, was defined as the shear angle under onset of buckling in this study. In addition, the shear angle was predicted up to the locking angle with pin-joint assumption and some geometrical parameters, such as yarn width, the space between yarns and friction, were considered for modifying the model. McBride and Chen¹⁷ dealt with change in internal geometry during shear deformation. They offered a geometric model for the unit cell of a woven fabric by considering four sinusoidal curves. In that model, it was deemed that the yarn width, yarn space and fabric height were measurable during the deformation. So, some input data for the model were to be measured beforehand. They found that the thickness of the fabric did not depend on shear deformation. For small shear deformation, the fiber volume fraction was found to increase directly with shear angle. Wang et al.¹⁸ reported that in a bias extension test, the aspect ratio of the specimen significantly affects the deformation pattern and gross stress–strain relation. In this study, the slippage is observed in carbon fabric during bias deformation. However, for glass fabric with a much lower bending rigidity, no clear slippage is observed under bias stress.

Page and Wang^19,20 dealt with the prediction of shear force and yarn slippage analysis by using the three-dimensional (3D) non-linear FEM. Zhang and Fu^21,22 and Zhang and Xu²³ suggested a two-dimensional (2D) micromechanical model for woven fabric and its application to analyze the buckling of fabric under uniaxial tension. Lo and Hu²⁴ tried to evaluate shear properties of fabric in all directions on the basis of Kilby’s model.⁴ They found that there is a strongly linear relationship between shear rigidity and shear hysteresis in all directions. Kuwazuru and Yoshikawa^25–27 studied anisotropy in tensile properties of plain weave fabric in an arbitrary direction numerically by a new concept of the Pseudo-continuum model. This model is constructed on the basis of three modes of fabric deformation and Strut-Spring concept.

Despite the cited investigations, it seems that the current knowledge is not adequate to evaluate geometric behavior and mechanical response of a woven fabric simultaneously on the basis of real deformation of yarns. Accordingly, this study addresses yarn deformation and yarn configuration in the structure of woven fabric when it is subjected to stress in an arbitrary direction. Consequently, the main effort of this paper would be on finding an approach to evaluate anisotropy of the geometrical and mechanical properties of a plain weave fabric. The required theory is discussed in this part of the paper and the authority of the proposed model and utilized empirical methods are presented in part II.

Deflection curve of tensioned yarn

Figure 1 indicates the path of a yarn in a unit cell of plain weave. The half path of yarn between two cross yarns is adequate to study the deflection curve of the yarn due to the symmetry of the path in the ideal structure.

Figure 1.

The yarn path between two cross yarns in fabric cross-section (a). Curvature and cantilever bending at a fixed point (b).

The well-known Euler–Bernoulli beam equation (1) is usually used in continuum mechanics for isotropic materials considering many assumptions. This equation reveals the relation between beam curvature and moment at a certain point of (x,y) by considering the EI quantity, where EI shows the stiffness of the beam, which consists of E, Young's modulus of the beam, and I, the moment of inertia of the beam cross-section:

\frac{1}{ρ_{(x, y)}} = \frac{M_{(x, y)}}{EI}

(1)

To derive Equation (1) it is assumed that the absolute values of deformation of materials in compressive and tensile stress are the same and that Hooke’s law is governed. However, there are some restrictions for applying this equation to evaluate the bending stiffness of textile materials, as well as yarn, due to the non-isotropic behavior of textile materials. Therefore, the behavior of textile materials is not the same as continuum materials when they are subjected to the bending moment.

It is clear that the response of yarns in axial compressive and tensile stress is dissimilar. Thus, considering EI cannot be used as yarn stiffness. Nevertheless, the yarn curvature at a certain point (x,y) is logically proportional to the ratio of the bending moment and yarn stiffness at this point. Therefore, the quantities EI do not exert directly in this study. Hereafter, we indicate the stiffness of yarn at point (x,y) as S₍ _x _, _y ₎, which will be estimated empirically. Thus, Equation (1) can be rewritten as Equation (2):

\frac{1}{ρ_{(x, y)}} = \frac{M_{(x, y)}}{S_{(x, y)}}

(2)

Moreover, the following assumptions are considered in this model.

Assumptions

Effects of yarn tension, yarn deformation and yarn flattening on yarn stiffness are negligible.

Yarns are fully elastic. In other words, the residual forces in the structure of bent yarn are disesteemed.

The yarn geometry and properties throughout the entire yarn are identical.

The yarn slippage at the intersection point is negligible.

The moment at a certain point can be computed by means of Equation (3) (Figure 1). On the basis of mathematical relations, Equation (4) depicts the relationship between curvature and yarn path function. Equation (5) can be deducted by substituting Equations (3) and (4) into Equation (2):

M_{(x, y)} = F_{n} (x - \frac{l}{2}) + T (y - h)

(3)

\frac{1}{ρ_{(x, y)}} = \frac{| y^{″} |}{{(1 + {(y^{'})}^{2})}^{\frac{3}{2}}}

(4)

\frac{y^{″}}{{(1 + {(y^{'})}^{2})}^{\frac{3}{2}}} = \frac{F_{n}}{S} (x - \frac{l}{2}) + \frac{T}{S} (y - h) y (0) = 0, y^{'} (0) = 0

(5)

The differential equation (5) has been analytically solved under stipulation that T > 2F_n by assuming y′ = 0. On the basis of the Experimental simulation section, when a planar force is larger than normal force (T > 2F_n), then Equations (6) would be the solution of Equation (5). Otherwise, when T ≤ 2F_n, a numerical solution must be considered for Equation (5). The authenticity of the analytical solution in comparison with the numerical solution in different conditions has been surveyed in the Experimental simulation section:

\begin{matrix} \begin{matrix} y = \frac{F_{n}}{2 T} e^{x \sqrt{\frac{T}{S}}} (- \frac{l}{2} - \frac{hT}{F_{n}} + \sqrt{\frac{S}{T}}) + \frac{F_{n}}{2 T} (e^{- x \sqrt{\frac{T}{S}}} (- \frac{l}{2} - \frac{hT}{F_{n}} - \sqrt{\frac{S}{T}}) - 2 x + l) + h \end{matrix} & If T > 2 F_{n} \end{matrix}

(6)

Where

h = \frac{F_{n} (e^{\frac{l}{2} \sqrt{\frac{T}{S}}} (- \frac{l}{2} + \sqrt{\frac{S}{T}}) + e^{- \frac{l}{2} \sqrt{\frac{T}{S}}} (- \frac{l}{2} - \sqrt{\frac{S}{T}}))}{T (e^{\frac{l}{2} \sqrt{\frac{T}{S}}} + e^{- \frac{l}{2} \sqrt{\frac{T}{S}}})}

(7)

Flattening of yarn

A compressive force on yarn causes flattening of the cross-section. Deformation of yarn cross-section is a function of fiber arrangement in the yarn structure, quantity and modality of compressive force, yarn tension, etc. However, in this study it is assumed that the flattening coefficient, namely e_d (Equation (8)), is only function of compressive force F_n. The relationship between d (the thickness of yarn under a certain compressive force) and compressive force (Equation (10)) can be determined via the empirical method, which is expressed in part II:

e_{d} = \frac{d - d_{0}}{d_{0}}

(8)

where d₀, the original diameter of the yarn, can be calculated for a certain value of yarn packing density and fiber density by Equation (9):

d_{0} = \sqrt{\frac{4 \cdot Tex}{10^{9} \cdot ρ \cdot π \cdot μ}}

(9)

e_{d} = G (F_{n})

(10)

Description of the model

The following classification can be contemplated in anisotropy of the tensile properties of fabric.

Remark: the subscripts 1 and 2 refer to warp and weft, respectively.

Relaxed fabric state

In this condition, it is deemed that the planar forces T₁ and T₂ are zero, then the differential equations (5) for both sides of the warp and weft yarns convert to Equations (11) and (12). These equations should be solved in a numerical manner due to the absence of planar force. Obviously, the interaction normal force F_n at the interlacing point is unique for both warp and weft yarns:

\frac{y_{1}^{″}}{{(1 + {(y_{1}^{'})}^{2})}^{\frac{3}{2}}} = \frac{F_{n}}{S_{1}} (x - \frac{l_{1}}{2}) y_{1} (0) = 0, y_{1}^{'} (0) = 0

(11)

\frac{y_{2}^{″}}{{(1 + {(y_{2}^{'})}^{2})}^{\frac{3}{2}}} = \frac{F_{n}}{S_{2}} (x - \frac{l_{2}}{2}) y_{2} (0) = 0, y_{2}^{'} (0) = 0

(12)

The quantities l₁ and l₂ (the space between two neighbor yarns) are computed on the basis of relations (13) and (14):

l_{1} = (1 + e_{1}) \cdot \frac{1}{D_{2}}

(13)

l_{2} = (1 + e_{2}) \cdot \frac{1}{D_{1}}

(14)

where D₁, D₂ and e₁, e₂ are the yarn sett (as known values) and fabric strain in the relative warp or weft direction, respectively. Evidently, in a relaxed condition, fabric strain in the warp and weft directions is zero (e₁ = e₂ = 0). On the other hand, the flattening functions for warp and weft can be written as in Equations (15) and (16) separately. These relations empirically reveal the flattening deformation of yarns as a function of normal force (F_n):

e_{d 1} = G_{1} (F_{n})

(15)

e_{d 2} = G_{2} (F_{n})

(16)

On the basis of fabric geometry, Equation (17) is always true:

h_{1} + h_{2} = \frac{d_{1} + d_{2}}{2}

(17)

where d₁ and d₂ are the thickness of the warp and weft yarns. Consequently, equilibrium f₁ can be deducted by substituting Equation (8) into Equation (17):

f_{1} = (d_{01} (1 + e_{d 1}) + d_{02} (1 + e_{d 2})) / 2 - (h_{1} + h_{2}) = 0

(18)

On the other hand, the length of the warp and weft yarns between two cross yarns can be calculated by relations (19) and (20), respectively:

L_{1} = \frac{C_{1} / 100 + 1}{D_{2}}

(19)

L_{2} = \frac{C_{2} / 100 + 1}{D_{1}}

(20)

where C₁ and C₂ are warp and weft yarn crimps, respectively (as known values). Indeed, the length of yarn between two cross yarns can be calculated via the deflection curve. Therefore, Equations (21) and (22) can be applied when the deflection curves of the warp and weft are determined:

f_{2} = 2 \int_{0}^{l_{1} / 2} \sqrt{1 + y_{1}} - L_{1} = 0

(21)

f_{3} = 2 \int_{0}^{l_{1} / 2} \sqrt{1 + y_{2}} - L_{2} = 0

(22)

Finally, a system of equations consisting of Equations (18), (21) and (22) can be contemplated to evaluate the relaxed fabric state. Three unknown values, F_n, S₁ and S₂, can be computed from this system of equations numerically by utilizing mathematical methods. Accordingly, a script has been constructed in the MATLAB program.

On the basis of assumptions, the stiffness of yarns, which are derived by the mentioned method in the relaxed fabric state, is constant during deformation. Hence, warp and weft yarns stiffnesses are deemed as known values in the following states.

Stress in the principal direction

Suppose that the direction of imposed stress is in the warp direction. Then, the planar force in cross yarns, here the weft yarn, is negligible. In this condition T₂ is zero, while both e₁ and e₂ are subjected to change. Evidently, quantity e₁ indicates the strain of fabric in the warp direction and e₂ shows the contraction of fabric in the weft direction. Equations (13) and (14) can be used to calculate new values of l₁ and l₂. Equation (23) is valid in the warp direction when the planar force in warp direction T₁ is larger than 2F_n. However, to indicate the deflection curve of the cross yarn (weft yarn), Equation (25) must be used:

\begin{matrix} y_{1} = \frac{F_{n}}{2 T_{1}} e^{x \sqrt{\frac{T_{1}}{S_{1}}}} (- \frac{l_{1}}{2} - \frac{h_{1} T_{1}}{F_{n}} + \sqrt{\frac{S_{1}}{T_{1}}}) \\ + \frac{F_{n}}{2 T_{1}} (e^{- x \sqrt{\frac{T_{1}}{S_{1}}}} (- \frac{l_{1}}{2} - \frac{h_{1} T_{1}}{F_{n}} - \sqrt{\frac{S_{1}}{T_{1}}}) - 2 x + l_{1}) + h_{1} \end{matrix}

(23)

Where

h_{1} = \frac{F_{n} (e^{\frac{l_{1}}{2} \sqrt{\frac{T_{1}}{S_{1}}}} (- \frac{l_{1}}{2} + \sqrt{\frac{S_{1}}{T_{1}}}) + e^{- \frac{l_{1}}{2} \sqrt{\frac{T_{1}}{S_{1}}}} (- \frac{l_{1}}{2} - \sqrt{\frac{S_{1}}{T_{1}}}))}{T_{1} (e^{\frac{l_{1}}{2} \sqrt{\frac{T_{1}}{S_{1}}}} + e^{- \frac{l_{1}}{2} \sqrt{\frac{T_{1}}{S_{1}}}})}

(24)

\frac{y_{2}^{″}}{{(1 + {(y_{2}^{'})}^{2})}^{\frac{3}{2}}} = \frac{F_{n}}{S_{2}} (x - \frac{l_{2}}{2}) y_{2} (0) = 0, y_{2}^{'} (0) = 0

(25)

Apparently, maximum yarn strain ε_1y can occur in the inflection point due to the large amount of tension at this point. Equation (26) can be utilized to estimate the strain of yarn, where as input data the inflection angle of the warp yarn is deducted from relation (27) and E₁ is the original warp yarn modulus. The maximum yarn strain quantity is considered for the whole yarn strain. Thus, Equation (21) must be modified to Equation (28). While the strain in cross yarns ε_2y is deemed negligible due to absence of T₂, then Equation (22) is still valid in this state:

ɛ_{y 1} = \frac{T_{1}}{E_{1} \cos β_{1}}

(26)

where

\tan β_{1} = y_{1}^{'} (l_{1} / 2)

(27)

f_{2} = 2 \int_{0}^{l_{1} / 2} \sqrt{1 + y_{1}^{2}} - L_{1} (1 + ɛ_{1 y}) = 0

(28)

Now, the system of equations is composed of Equations (18), (22) and (28). This system of equations can be solved for certain value of T₁. Eventually, the variables F_n, e₁ and e₂ can be estimated by this system of equations. When the fabric is suffering the stress in the weft direction, then the proposed equations can be utilized by substituting relative subscripts (1 by 2) in the same way (Equations (29)–(34)).

If T₂ > F_n then:

\begin{matrix} y_{2} = \frac{F_{n}}{2 T_{2}} e^{x \sqrt{\frac{T_{2}}{S_{2}}}} (- \frac{l_{2}}{2} - \frac{h_{2} T_{2}}{F_{n}} + \sqrt{\frac{S_{2}}{T_{2}}}) \\ + \frac{F_{n}}{2 T_{2}} (e^{- x \sqrt{\frac{T_{2}}{S_{2}}}} (- \frac{l_{2}}{2} - \frac{h_{2} T_{2}}{F_{n}} - \sqrt{\frac{S_{2}}{T_{2}}}) - 2 x + l_{2}) + h_{2} \end{matrix}

(29)

Where

h_{2} = \frac{F_{n} (e^{\frac{l_{2}}{2} \sqrt{\frac{T_{2}}{S_{2}}}} (- \frac{l_{2}}{2} + \sqrt{\frac{S_{2}}{T_{2}}}) + e^{- \frac{l_{2}}{2} \sqrt{\frac{T_{2}}{S_{2}}}} (- \frac{l_{2}}{2} - \sqrt{\frac{S_{2}}{T_{2}}}))}{T_{2} (e^{\frac{l_{2}}{2} \sqrt{\frac{T_{2}}{S_{2}}}} + e^{- \frac{l_{2}}{2} \sqrt{\frac{T_{2}}{S_{2}}}})}

(30)

\frac{y_{1}^{″}}{{(1 + {(y_{1}^{'})}^{2})}^{\frac{3}{2}}} = \frac{F_{n}}{S_{1}} (x - \frac{l_{1}}{2}) y_{1} (0) = 0, y_{1}^{'} (0) = 0

(31)

ɛ_{y 2} = \frac{T_{2}}{E_{2} \cos β_{2}}

(32)

where

\tan β_{2} = y_{2}^{'} (l_{2} / 2)

(33)

f_{2} = 2 \int_{0}^{l_{1 / 2}} \sqrt{1 + y_{2}^{2}} - L_{2} (1 + ɛ_{2 y}) = 0

(34)

Stress in arbitrary direction, except of principal directions

When a strip fabric is subjected to stress in an arbitrary direction, except the principal direction, all quantities e₁, e₂, Ψ₁, Ψ₂ and F_n are subjected to change. The current equations are not adequate to achieve the final solution. It seems more assumptions must be contemplated in this regard. For this purpose, the exterior geometry of the fabric is discussed.

Exterior geometry of plain weave fabric after deformation in an arbitrary direction

Figure 2(a) describes the deformation of a specimen when the initial angle of the force axis with weft yarns is φ. Suppose one imaginary square abcd on the center of the specimen, which is parallel to the edges of the strip sample. Square abcd is converted to parallelogram a′b′c′d′ after deformation. Figure 2(b) illustrates the deformation of square abcd in detail. On the basis of observations, the following assumptions are deemed.

Figure 2.

Imaginary square abcd in the center and parallel with the sample’s edges. (a) Deformation of square. (b) Converting the square to a parallelogram in detail.

Assumptions

Deformation is symmetric around the central point of specimen ‘O’.

The lateral sides of the imaginary parallelogram are parallel to the force axis.

The deformation and distribution of force are homogonous for each yarn and the cut-end effect is negligible.

The perimeter of the imaginary parallelogram is constant during deformation and is the same as the initial imaginary square abcd before deformation.

The jaws effect on deformation on the area under consideration is negligible.

On the basis of pure geometry and assumptions, subsequent arguments are valid. Lateral sides of the parallelogram are parallel to the force axis, thus:

e_{1} = \frac{(1 + e_{2}) \cdot \sin Ψ_{2}}{\tan ϕ \cdot \sin Ψ_{1}} - 1

(35)

The strain of the fabric can be calculated as Equation (36):

e_{f} = (1 + e_{1}) \sin ϕ \cdot \cos Ψ_{1} + (1 + e_{2}) \cos Ψ_{2} \cdot \cos ϕ - 1

(36)

Remark: the fabric strain calculated by Equation (36) is not the same as the fabric strain recorded by Tensile Instron, as this equation computes the regional strain of fabric in the direction of the force axis.

The oblique angle of the parallelogram can be found by Equation (37):

\tan α = \frac{(1 + e_{1}) \cdot \cos Ψ_{1} - \sin ϕ - e_{f} \cdot \cos ϕ}{(1 + e_{1}) \cdot \sin Ψ_{1}}

(37)

The equilibrium (38) is logically true when it is deemed that the perimeter of the parallelogram is constant:

f_{4} = \frac{2 (1 + e_{1}) \cdot \sin Ψ_{1}}{\cos α} + 2 \cos ϕ \cdot (1 + e_{f}) - 4 \cos ϕ = 0

(38)

Distribution of planar force

Figure 3 indicates typically the applied force on the weft yarn system of a strip woven fabric under certain force F_y, where F_y is recorded by the load cell of the tensile machine. The contribution of each system of yarns to reaction is a function of fabric density and the angle of each system of yarns with the tension axis. Equations (39) and (40) demonstrate these relations for warp and weft yarns, respectively:

T_{1} = F_{y} \frac{\sin ϕ}{D_{1} \cdot ω \cdot \cos Ψ_{1}}

(39)

T_{2} = F_{y} \frac{\cos ϕ}{D_{2} \cdot ω \cdot \cos Ψ_{2}}

(40)

Figure 3.

Contribution of the weft yarn system to bearing applied force.

Solving system of equations

When a fabric is subjected to stress in an arbitrary direction, except principal directions, the planar forces T₁ and T₂ are larger than F_n after a bit strain. On the basis of previous arguments, the yarn deflection Equation (6) can be applied in both warp and weft yarns by using relative subscripts (Equations (23) and (29)). In this case, there are warp and weft yarns strain due to the existence of planar force in both the warp and weft directions. The system of equations includes five equations, (18), (28), (34), (35) and (38), and can be applied to determine five variables (Table 1) for a certain value of F_y.

Table 1.

Input data and variables

Input data	Variables
C₁, C₂	e₁, e₂
d₀₁, d₀₂	Ψ₁, Ψ₂
D₁, D₂	F_n
E₁, E₂
G₁ (F_n), G₂ (F_n)

Table 1 illustrates the inputs and variables of this model when the stress is imposed in an arbitrary direction, except principal directions. According to the stated arguments, a script has been constructed in the MATLAB program to solve equations. This script reports all corresponded values, including fabric geometry and fabric strain for a given value of F_y, until one of the yarn systems reaches ultimate strain (i.e. $ɛ_{y 1} = ɛ_{y 1 u}$ and/or $ɛ_{y 2} = ɛ_{y 2 u}$ ).

Experimental simulation

To solve deflection Equation (5), two numerical and analytical methods are compared in this section. Accuracy of the numerical method is higher than that the analytical solution, Equation (6), as the latter method is obtained by assuming first-order derivation quantity (y′) as zero. However, the numerical solution method is time consuming in contrast with the analytical method. Figure 4 indicates the analytical and numerical solutions when the planar force is zero. It can be observed that the analytical method is not trustworthy when the planar force is too small or is negligible compared to normal force. Accordingly, in the relaxation condition (Relaxed fabric state section) and when the stress is imposed in the principal direction (Stress in the principal direction section), the behavior of yarn with zero planar force has to be assessed using a numerical method.

Figure 4.

Analytical (—) and numerical (…) solution of deflection equation for S = 1E-9[N·m²]: planar force is zero and normal force increases gradually.

Figure 5 compares the numerical and analytical solution for deflection of a yarn with 1 E-9 [N·m²] stiffness and 2.33 E-4 [m] as l/2 in different planar force to normal force ratios (T/F_n), namely 10, 5 and 2, respectively. It can be argued that Equation (6) is reliable when the planar force to normal force ratio is 2 or bigger. Different conditions for verifying this method can be found in Dolatabadi.²⁸

Figure 5.

Analytical (—) and numerical (…) solution of deflection equation for S = 1E-9[N·m²]: planar force increases gradually with proportions 10, 5 and 2.

Conclusion

Many factors can affect the behavior of woven fabric in an arbitrary direction. The current work is an effort to characterize some of these factors as a comprehensive model, including the geometrical and mechanical behavior of a fabric. The mechanical and geometrical behaviors of a fabric under stress in principal directions and arbitrary directions are studied as part I in this paper. Part II will deal with verifying the proposed model and explaining the test methods. The yarn rotation at the intersection and crimp-interchange is a dominant phenomenon when there is low amount of deformation in the off-axis directions, while the tensile deformation mostly occurred when there was a large amount of deformation. Indeed, it is difficult to determine at which stress the shear mode of deformation is transferred to tensile deformation.

Footnotes

Funding

This work was supported by the Grant Agency of the Czech Republic (No. 106/09/1916 of GACR).

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Anisotropy in tensile properties of plain weave fabric – Part I: The meso-scale model

Abstract

Keywords

Introduction

Deflection curve of tensioned yarn

Assumptions

Flattening of yarn

Description of the model

Relaxed fabric state

Stress in the principal direction

Stress in arbitrary direction, except of principal directions

Exterior geometry of plain weave fabric after deformation in an arbitrary direction

Assumptions

Distribution of planar force

Solving system of equations

Experimental simulation

Conclusion

Footnotes

Funding

References